Re: a hopeful request

Thanks, Bill! For the record and those who are not familar with the details of 2-descent, the map quartic->E has degree 4 so there are up to 4 preimages of a rational point on E. More precisely, if there is at least one preimage, then the number is equal to the size of 2-torsion in E(Q). (For Randall's curves the 2-torsion is all rational so there will be 4 preimages if any.) This is just like the situation for the multiplcation-by-2 map from E to itself (and for good reason, since if the quartic curve has any rational points then it is isomorphic to E and the map Bill implemented is just multplication by 2 in disguised form).

John

On Sat, 2 Mar 2024 at 17:03, Bill Allombert <Bill.Allombert@math.u-bordeaux.fr> wrote:

On Fri, Mar 01, 2024 at 04:56:23PM -0800, American Citizen wrote:
> To all:
>
> I have been looking for this for a long time.
>
> Suppose I have an elliptic curve in Weierstrass format [a1,a2,a3,a4,a6] and
> a two-covering or quartic a*x^4 + b*x^3 + c*x^2 + d*x + e and they share the
> same invariants, the I invariant and the J invariant.
>
> Is it possible to develop the forward and inverse maps for points on either
> the elliptic curve mapping to the quartic two-cover, or the points on the
> two-cover mapping back to the elliptic curve?
>
> I would like to obtain the forward and inverse maps for points on either the
> elliptic curve -> quartic or quartic -> elliptic curve

Please find the following GP script with a function
quartic_to_ellmap(Q) that given a quartic y^2=q(x), returns
[E,f,g] where E is an elliptic curve, f a map from q to E and
g a (one-to-many) map from E to q.

There is a test.

Cheers,
Bill.